Random mixture of competing XY and Ising anisotropies

Volume 122, number 6,7 PHYSICS LETTERS A 22 June 1987 RANDOM MIXTURE OF COMPETING XV AND ISING ANISOTROPIES Rita M. ZORZENON DOS SANTOSa,!, A.M. MARIZ~’and Raimundo R. DOS SANTOS~ Departamento de Fisica, Pontificia Universidade Católica, 22453 Rio de Janeiro RJ, Brazil Departamento de Fisica, Universidade Federal do Rio Grande do Norte, 59000 NatalRN, Brazil Received 20 February 1987; accepted for publication 1 April 1987 Communicated by A.A. Maradudin We study a ferromagnetic system in which each bond can have either 0(1) (Ising) or 0(2) (XY) orthogonal symmetries at random with concentrations I —p and p, respectively. By means of a Migdal—Kadanoff approximation we obtain the global phase diagram in the temperature—concentration—anisotropy space. Sections ofthe phase diagram are obtained in which the boundaries between paramagnetic and ordered phases display sharp minima consistent with experimental and c-expansion predictions. Systems with competing random anisotropies have been the subject of several recent experimental investigations (see, for instance, refs. [1—3]).In these systems one kind of magnetic atom is replaced by another in such a way that the otherwise pure systems have orthogonal order parameters. For instance , ~,,Co,,Cl2 (see ref. [1]) and, ~Co~Br2(see ref. [2]) order along the crystal c-axis or inTheoretical the c-plane when p is either 0 or 1, respectively. studies of general random anisotropies [4] indicate the existence of three ordered phases that meet at a (decoupled) tetracritical point: the two ordinary pure phases and a mixed phase in which there is simultaneous ordering of both components of the mixture, For some random XY—Ising mixtures [1,2], however, the low-temperature transition between “pure” and “mixed” phases is smeared; this fact was mitially attributed to random-field effects [21, but offdiagonal couplings can also give rise to a smeared transition [5]. The purpose of this work is to study the global phase diagram in the temperature—concentration—anisotropy space of a random XY—Ising mixture within a position space renormalisation group (PSRG) framework. In this way, one expects to display theoretical information complementary to earPresent address: Instituto de Fisica, Universidade Federal Fluminense, 24000 Niterói RJ, Brazil. her c-expansion studies [4,51. We consider here a random-bond XY—Ising mixture described by the following hamiltonian (in units of kT) — ~ K1~[(1 4~)( a~a~ + a~’cr~’) 2J (1) + (1+4,‘ )~a where K 1~is the exchange coupling between nearest neighbour pairs <if> of spins on a cubic lattice; A~ is the anisotropy (in spin-space) parameter and the a are Pauli matrices. On universality grounds one does not expect qualitative differences between random-bond and random-site mixtures. We assume that the coupling K,, and anisotropy 4,, are governed by the distribution p(K 4 [I’! ~4 ‘ ii’ 0/ Pi — — — — ‘. +pô(40 —4)]ö(K~~ —K0). (2) A few remarks about this distribution are in order. First, we assume for simplicity that the actual bond strength K0 is always the same, irrespective of the anisotropy 4,,, (i.e., irrespective of the magnetic species present). Secondly, as we vary A we cover XY—Ising (—1 ~4 <0) and Heisenberg-.Ising (4=0) mixtures, the former being competing but not the latter [61; for 4>0 one has essentially a mixture of 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) — 367 Volume 122, number 6,7 PHYSICS LETTERS A b=r 2 and d= 3 one has four branches whose K~5~ simply add LK(s) +K~ +K~ +K~ (4a) Ising and anisotropic (Ising-hike) Heisenberg bonds in which the randomness plays no significant role [6]. Also, by allowing 4 to vary, one guarantees that the parameter space is invariant under renormahisation group (RG) iterations, As pointed out previously [61 one of the most efficient PSRG schemes amenable to treat random , and the resulting anisotropy is given by 2’ 4, K~,4,,) J~)~ 4, K,,, Ak) Now, g and 2’ are functions of eight pairs (K~,4), each of which is distributed according to eq. (2). Therefore, after each RG iteration one has to examinc the renormalised distribution, as given by ref. [11] P(k (8 1~41)=jfl (5) which is no longer binary and gets more complicated as the number of RG iterations increase. To avoid this uncontrollable proliferation of d-functions one usually resorts to a truncation scheme which consists of keeping the distribution binary [121, i.e., (3b) , [dK~ d4 P(K,,,4)] jI (3a) , (4b) K~s)4,ls). ~ 1—•- I quantum systems in three dimensions is the Migdal—Kadanoff (MK) approximation [7,8] which consists essentially of 1 D decimation followed by bond-moving. The decimation operation is carried out following the scheme of Caride et al. [9,10] to yield, for a scaling factor b= 2, the following functions (from now on, the bonds within the MK clusters will be labelled by a single index) 5~ ~K~(K K~ 1, 22 June 1987 which are to be calculated numerically; this result can be thought of as a “series” combination ofbonds 4) and (Kk, Ak). Accordingly, the bond-moving operation can be regarded as a “parallel” combination of the decimated bonds (K~,4~s)) so that for ~ ~1 C E XY H ‘I Ii I A III \ I ~ / I I I I, / / / I I , I I I ‘ B i ‘.‘ / / A .-“.-‘‘—, I IC I II I I I I I” ,‘ / / / / / / -a~z I-p Fig. I. Critical surface (schematic) in the (kT/J, 4, l—p) space obtained for a competing XY—Ising mixture. Critical fixed points are denoted by (*), and zero temperature attractors by (U). 368 Volume 122, number 6,7 P’(K,, 4,) [(1 —p’)ö(A, PHYSICS LETTERS A — 1) points and the critical surface can be obtained by iterating the RGT, as usual [14]. The resulting phase diagram in the T—p—4 space is shown in fig. 1. Note that the fixed point and exponents are the same as those obtained in ref. [8] (see also ref. [15]); when p = 0 one has an Ising model so that the critical line (6) +p’c5(A~—4’)]o(k’, —K’) with p’, A’ and K’ chosen in such a way that the following averages are preserved , (7a) <K,(1 —A,) >,~.=<K,(l —4,) >~. <K~(l—4~)>,~=(K~(l—4~) >,~. lies at l/K~=7.66forany4. The most important feature of the high-temperature transitions between paramagnetic and ordered phases is presented in fig. 1 (see also fig. 2): for small concentrations of either component of the mixture, the behaviour of the critical temperature is similar to ordinary dilution [16]. To understand this, consider an XY-rich mixture (i.e. p ~ 1). Since XY correlations cannot propagate through “orthogonal” Ising bonds, the latter then behave as if they were absent. A similar analysis holds for the Ising-rich regime (p ~ 0). In the intermediate concentration (7b) , (7c) , 22 June 1987 where the subscripts refer to either (5) or (6). There is certainly a degree of arbitrariness in the choice (7); other choices were tried [13] without major qualitative changes in the phase diagrams. The RG transformation (RGT), defined in terms of p’, K’ and A’ as above, can be solved for fixed keIC —a8.0 70 • -1.0 -0.8 .~ 6.0 • •.• ~ 5.0 ~.,, ~ , ‘‘.. ..•• \N “. ‘. \ ‘N N~ ~ 40 N N, \\ 30 ~ N // •>, ‘K ~ ~~\/~>\/ Y ‘ I 0.25 I 0.50 -0.2 .~ \,,,~‘ j j I ~\ ~I J .0 0.0’ 0.0 -0.4 2:’.. ‘I \ .7 7’ ~ ~, 7’ Z• ~‘ ‘ \ 2.0 “.• -0.6 7~ ,‘ I 0.75 1.0 P Fig. 2. Sections of fig. I for the critical temperature (kTIJ) as a function of the concentration ofXYbonds (p) for several values of the anisotropy parameter (4) as labelled on the right-hand side. 369 Volume 122, number 6,7 PHYSICS LETTERS A range, however, competition between the two species sets in, marked by a pronounced minimum in the phase diagram (line AH in fig. 1). From the RG flow directions, we can extract three distinct critical behaviours for the finite temperature transitions (see fig. 1): (i) XY critical behaviour, for points lying on the surface bounded by AEH; (ii) Ising critical behaviour, for points on the surface bounded by ABCIDH; (iii) Heisenberg critical behaviour, for points on the line AR. Although this latter behaviour should be taken with care, one must have in mind that the intermediate region involves the simultaneous ordering of two order parameters (three components, on the whole). Experimental results for Fe1 ~Co~CI2 and Fe1 ~Co~Br2yield a tetracritical concentration (i.e., the one at which the three phases meet, for a given 4), p1~0.3,which is below any of ours (see fig. 2). Several aspects, that were not included in our model, can account for this discrepancy. Firstly, our original distribution, eq. (2), only distinguishes between Ising (4 = 1) and non-Ising (4 ~ 1) bonds as outcomes of RG iterations (see eq. (6)). Secondly, geometrical aspects are not fully present, since one does not take into account the presence of XY- and Ising-percolating clusters more explicitly. Finally, neither random-field effects nor off-diagonal couplings were considered. Unfortunately, all these features cannot be considered without significantly enlarging the parameter space which, in turn, would involve a great deal more of computer time due to the configurational averaging. These points will be separately investigated in the future. Our description of the low temperature region, however, is not satisfactory. Firstly, there is no XY zero-temperature attractor which is reflected in the XY ordered phase collapsing into the pure Heisenberg fixed point; this feature is present in the nonrandom case and is attributed to the lack of XY ~ metry in the two-spin cluster obtained under decimation of a three-spin cluster [13,171. Secondly, there is no intermediate mixed phase: the Ising-rich changes into XY-rich rather abruptly, as can be seen from fig. 2; the reason for this is similar to the lack of 0(2) symmetry in the decimated two-spin cluster, since the mixed phase should appear as a simultaneous ordering of several components of an order parameter. Also, one cannot precise whether the bor- 370 22 June 1987 der between the Ising and XY low temperature phases is a critical surface or a coexistence region. To conclude, one can say that the high temperature transitions between paramagnetic and ordered phases are well described by our approach. For the low temperature region, one needs an RG scheme that picks up the subtleties involved in both the pure XY and mixed phase orderings without losing its simplicity. Results for the competing Ising—Ising mixture [131 will appear elsewhere. The authors would like to thank P.M. Ohiveira, S.L.A. de Queiroz and C. Tsallis for helpful discussions and suggestions. One of us (RRS) is grateful to Professor K. Katsumata for sending several preprints on this subject prior to publication. Financial support from the Brazilian Agencies FINEP, CNPq and CAPES is also gratefully acknowledged. References [I] K. Katsumala, J. Tuchendler and S. Legrand, Phys. Rev. B 30 (1984) 1377. [2] P. Wong, P.M. Horn, R.J. Birgeneau and G. Shirane, Phys. Rev. B 27 (1983) 428. 131 K. Katsumata, M. Kobayashi, T. SatoandY. Miyako, Phys. Rev. B 19(1979)2700. [41S. Fishman and A. Aharony, Phys. Rev. B 18 (1978) 3507. [51M. Oku and H. Igarashi, Prog. Theor. Phys. 70 (1983) 1493. [61R.M. Zorzenon dos Santos, A.M. Mariz, R.R. dos Santos and C. Tsallis, J. Phys. C 18 (1985) 5475; A.M. Mariz and C. Tsallis, Phys. Rev. B 31(1985) 7491. [71A.A. Migdal, Soy. Phys. JETP 42 (1976) 743; L.P. Kadanofl’~Ann. Phys. (NY) 100 (1976) 559. [81H. Takano and M. Suzuki, J. Stat. Phys. 26 (1981) 635. [91A.0. Caride, C. Tsallis and S. Zanette, Phys. Rev. Lett. 51 (1983) 145, 616. [10] AM. Mariz, C. Tsallis and A.0. Caride, J. Phys. C 18 (1985) [II] [12] [13] [14] [15] [16] [17] 4189. T.C. Lubensky, Phys. Rev. B 11(1975) 3573. RB. 3221.Stinchcombe and B.P. Watson, J. Phys. C 9 (1976) R.M. Zorzenon dos Santos, Ph.D. Thesis, PUC/RJ (1986), unpublished. K.G. Wilson and J. Kogut, Phys. Rep. C 12 (1974). A.M. Mariz, R.M. Zorzenon dos Santos, C. Tsallis and R.R. dos Santos, Phys. Lett. A 108 (1985) 95. RB. Stinchcmnbe, in: Phase transitions and critical phenomena, Vol. 7, eds. C. Domb and J. Lebowitz (Academic Press, New York, 1983). C. Castellani, C. di Castro and J. Ranninger, Nuci. Phys. B 200 (1982) 45.